An infinite semigroup S such
that every nontrivial homomorph of it is isomorphic to S is called an HI semigroup.
Every commutative HI semigroup is a group and thus it is isomorphic to the group
Z(p)∞, for some prime P. An infinite Brandt semigroup is HI if and only if it has a
trivial structure group. An inverse HI semigroup containing a primitive idempotent
is either Brandt or else it is isomorphic to a trasfinite chain of extensions of a Brandt
semigroup K by isomorphic copies of K (where K has the trivial group
as its structure group). Necessary and sufficient conditions are given for a
semigroup of the latter type to yield an HI semigroup and an example is
constructed.
